Equilibrium recombination temperature for Helium

[Chromatic_Universe]

How to get to the expression for equilibrium recombination temperature for Helium?

 

[AndyW]

Hello,

The equilibrium recombination temperature for helium can be derived using the Saha equation, which relates the ionization fraction of an element to its temperature and density. The Saha equation is given by:

n_i/n_{i+1} = (2πm_e k_B T/h^2)^{3/2} * (2U_{i+1}/U_i) * e^{-E_i/k_B T}

where n_i and n_{i+1} are the number densities of the element in the ionized and neutral states, respectively, m_e is the mass of an electron, k_B is the Boltzmann constant, T is the temperature, h is the Planck constant, U_i and U_{i+1} are the partition functions for the ionized and neutral states, and E_i is the ionization energy.

To get the equilibrium recombination temperature for helium, we need to set the ionization fraction to 1, since at equilibrium, all helium atoms will be in the neutral state. This gives us the equation:

1 = (2πm_e k_B T_{rec}/h^2)^{3/2} * (2U_{i+1}/U_i) * e^{-E_i/k_B T_{rec}}

where T_{rec} is the equilibrium recombination temperature.

To solve for T_{rec}, we can take the natural logarithm of both sides and rearrange the equation to get:

ln(1) = ln[(2πm_e k_B T_{rec}/h^2)^{3/2} * (2U_{i+1}/U_i) * e^{-E_i/k_B T_{rec}}]

Using logarithm rules, we can simplify this to:

0 = (3/2)ln(2πm_e k_B T_{rec}/h^2) + ln(2U_{i+1}/U_i) - E_i/k_B T_{rec}

Rearranging and solving for T_{rec}, we get the final expression for the equilibrium recombination temperature for helium:

T_{rec} = \frac{E_i}{ln(2U_{i+1}/U_i) - (3/2)ln(2πm_e k_B/h^2)}

I hope this helps! Let me know if you have any further questions.